
where \((\eta _x, \eta _r)>0\)  are two learning rates.
We retrieve a continuous time gradient flow when \((\eta _x, \eta _r)\rightarrow 0\) .
Using such a model \(_\theta \)  and such updates is proved in [1]} to be equivalent to an unbalanced gradient flow in the space \(()\) , in contrast with classical (balanced) Wasserstein flows optimizing over \(_1()\) .
The update on \(r_i\)  is called a mirror descent step, and is used to enforce that \(r_i\ge 0\) .
The parameterization of \(_\)  using \(r_i^2\)  and not \(r_i\)  improves the numerical stability, because for the latter parameterization, the mirror update reads
\(r_i^{(t+1)} & = r_i^{(t)}.\exp \Big ( - 2\tfrac{\eta _r}{r_i^{(t)}} \nabla _{r_i} (_\theta ^{(t)}, ) \Big ),\) 
